Finite Group DatabasebetaFinite Group Databasebeta| order | 7232 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C113 ⋊ #341 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22⋊#90673 |
| commutator | C4×C113 | C42 |
| Frattini | C2×C4 | C2×#90673 |
| Fitting | M16×C113 | C4 |
| socle | C2×C113 | #95 |
| hypercenter | M16 | #90673 |
| hypocenter | C113 | #341 |
| LCS | G ▷ C4×C113 ▷ C2×C113 ▷ C113 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ M16 | |
90 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,113|a^2=d,b^2=e,c^2=f,d^2=f,e^2=f,b^a=bc,c^a=cf,e^a=ef,g^b=g^98,g^e=g^112308 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#20299473, #28406267, #28406348, … |
| maximal normal subgroup of | C2×#20299473, #28406267, #28406348, … |
| maximal quotient of | #28405403, #28405405, #28405407, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7232, 1424);